Abstract
Suppose that a homogeneous linear differential equation has entire coefficients of finite order and a fundamental set of solutions each having zeros with finite exponent of convergence. Upper bounds are given for the number of zeros of these solutions in small discs in a neighbourhood of infinity.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alotaibi A.: On Bank-Laine functions. Comput. Methods Funct. Theory 9, 335–345 (2009)
Alotaibi A., Langley J.K.: On solutions of linear differential equations with entire coefficients. J. Math. Anal. Appl. 367, 451–460 (2010)
Bank S.B., Laine I.: On the oscillation theory of f′′ + Af = 0 where A is entire. Trans. Am. Math. Soc. 273, 351–363 (1982)
Bank S.B., Laine I.: Representations of solutions of periodic second order linear differential equations. J. Reine Angew. Math. 344, 1–21 (1983)
Bank S.B., Laine L.: On the zeros of meromorphic solutions of second-order linear differential equations. Comment. Math. Helv. 58, 656–677 (1983)
Bank S.B., Laine I., Langley J.K.: On the frequency of zeros of solutions of second order linear differential equations. Result. Math. 10, 8–24 (1986)
Bank S.B., Laine I., Langley J.K.: Oscillation results for solutions of linear differential equations in the complex domain. Result. Math. 16, 3–15 (1989)
Bank S.B., Langley J.K.: Oscillation theory for higher order linear differential equations with entire coefficients. Complex Var. Theory Appl. 16, 163–175 (1991)
Drasin, D., Langley, J.K.: Bank-Laine functions via quasiconformal surgery. Transcendental Dynamics and Complex Analysis. London Math. Soc. Lecture Notes, vol. 348, pp. 165–178. Cambridge University Press, Cambridge (2008)
ElZaidi, S.M.: Some theorems concerning linear differential equations in the complex domain. Ph.D. thesis, University of Nottingham (1996)
ElZaidi S.M.: On Bank-Laine sequences. Complex Var. Theory Appl. 38, 201–220 (1999)
Frank G., Hellerstein S.: On the meromorphic solutions of nonhomogeneous linear differential equations with polynomial coefficients. Proc. Lond. Math. Soc. 3(53), 407–428 (1986)
Hayman W.K.: Meromorphic Functions. Oxford Mathematical Monographs. Clarendon Press, Oxford (1964)
Kim W.J.: The Schwarzian derivative and multivalence. Pac. J. Math. 31, 717–724 (1969)
Laine I.: Nevanlinna Theory and Complex Differential Equations. de Gruyter Studies in Mathematics, vol. 15. Walter de Gruyter & Co., Berlin (1993)
Langley J.K.: Some oscillation theorems for higher order linear differential equations with entire coefficients of small growth. Result. Math. 20, 517–529 (1991)
Langley, J.K.: On entire solutions of linear differential equations with one dominant coefficient. Analysis 15, 187–204; Corrections in: Analysis 15, 433 (1995)
Langley J.K.: Bank-Laine functions with sparse zeros. Proc. Am. Math. Soc. 129, 1969–1978 (2001)
Rossi J.: Second order differential equations with transcendental coefficients. Proc. Am. Math. Soc. 97, 61–66 (1986)
Shen L.C.: Solution to a problem of S. Bank regarding the exponent of convergence of the solutions of a differential equation f′′ + Af = 0. Kexue Tongbao 30, 1581–1585 (1985)
Shen L.C.: Construction of a differential equation y′′ + Ay = 0 with solutions having prescribed zeros. Proc. Am. Math. Soc. 95, 544–546 (1985)
Author information
Authors and Affiliations
Corresponding author
Additional information
Both authors acknowledge with thanks technical and financial support from the Deanship of Scientific Research, King Abdulaziz University, Jeddah, grant no. (91/130/1431).
Rights and permissions
About this article
Cite this article
Alotaibi, A., Langley, J.K. The Separation of Zeros of Solutions of Higher Order Linear Differential Equations With Entire Coefficients. Results. Math. 63, 1365–1373 (2013). https://doi.org/10.1007/s00025-012-0273-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00025-012-0273-7